Submitted Paper

Inserted: 24 mar 2026
Last Updated: 18 apr 2026

Pages: 15
Year: 2026

Abstract:

In this paper we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schrödinger type operator $\mathcal{H}_V:=-\operatorname{div}(A\nabla)+V$, where $V$ is convex and Kato decomposable, using the trace class property of the generated semigroup. As a consequence, using the ultracontractivity of the semigroup we obtain the log-concavity of the ground state which is also strong log-concave under additional assumptions on $\Omega$ and $V$.

Keywords: Brunn-Minkowski inequality, Schr\"odinger operators, first Dirichlet eigenvalue, Trace class semigroups, Ultracontractivity estimates

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• BM_Schrodinger.pdf